We present the results of our observations of the early stages of the 2012–2013 outburst of the transient black hole X-ray binary (BHXRB), Swift J1745–26, with the Very Large Array, Submillimeter Array, and James Clerk Maxwell telescope (SCUBA–2). Our data mark the first multiple-band mm and sub-mm observations of a BHXRB. During our observations the system was in the hard accretion state producing a steady, compact jet. The unique combination of radio and mm/sub-mm data allows us to directly measure the spectral indices in and between the radio and mm/sub-mm regimes, including the first mm/sub-mm spectral index measured for a BHXRB. Spectral fitting revealed that both the mm (230 GHz) and sub-mm (350 GHz) measurements are consistent with extrapolations of an inverted power law from contemporaneous radio data (1–30 GHz). This indicates that, as standard jet models predict, a power law extending up to mm/sub-mm frequencies can adequately describe the spectrum, and suggests that the mechanism driving spectral inversion could be responsible for the high mm/sub-mm fluxes (compared to radio fluxes) observed in outbursting BHXRBs. While this power law is also consistent with contemporaneous optical data, the optical data could arise from either jet emission with a jet spectral break frequency of Vbreak ≳ 1 x 10^14 Hz or the combination of jet emission with a lower jet spectral break frequency of Vbreak ≳ 2 x 10^11 Hz and accretion disk emission. Our analysis solidifies the importance of the mm/sub-mm regime in bridging the crucial gap between radio and IR frequencies in the jet spectrum, and justifies the need to explore this regime further.